If in the expansion of (a + b)ⁿ and (a + b)^{n + 3}, the ratio of the coefficients of second and third terms and third and fourth terms respectively are equal then n is _____.

If in the expansion of (1 +x)ⁿ, the coefficients of fifth, sixth and seventh terms are inA.P. then n is equal to _____.

If in the expansion of (1 +x)¹⁵, the coefficient of (2x + 3)^th and (r - 1)^th terms are equal then r is equal to _____.

The coefficient of x⁵y⁸ in the expansion of (x + y)¹³ is _____.

Constant term in the expansion of \(\left( x-\frac { 1 }{ x } \right) ^{ 14 }\) _____.

Find the value of [x+\(\sqrt { x-1 } \)]⁶+[\(\sqrt { x-1 } \)]⁶ , using binomial theorem.

Find the middle term in the expansion of \(\left( \frac { x }{ a } -\frac { a }{ x } \right) ^{ 10 }\)

Evaluate the following by using Binomial theorem:
(105)³

Write the general term in the expansion of \(\left( 2x^{ 2 }+\frac { 1 }{ x } \right) ^{ 12 }\)

The 3^rd, 4^th and 5^th terms in the expansion of (x +a)ⁿ are 84, 280 and 560 respectively. Find the values of x, a and n.

Simplify (x + 2y)⁸ + (x - 2y)⁸

Find the coefficient of x5 in the expansion of (1 + 3x)⁶ (1 - X)⁵

Find the 4th term in the expansion of \(\left( 3x-\frac { { y }^{ 3 } }{ 6 } \right) ^{ 4 }\)

Find the coefficient of x6 in the expansion \(\left( x-\frac { { 1 } }{ 6^2 } \right) ^{ 24 }\) .

Find the term independent of x in the expansion of \(\left( 2x+\frac { 3 }{ { x }^{ 2 } } \right) ^{ 9 }\)

Find the coefficient of x⁵ in the expansion of (1 + x)³ (1 + x)⁶.

For what value of m, the coefficients of the (2m + 1)th and (4m + 5)th terms in the expansion of (1 + x)¹⁰ are equal?

Find a, b and n in the expansion of (a + b)ⁿ if the first three terms of the expansion are 729, 7290 and 30375 respectively.

If the fourth term in the expansion of \(\left( ax+\frac { 1 }{ x } \right) ^{ n }is \frac { 20 }{ 27 } \) then find the value of a and n.